318 Chapter 7 The Normal Distribution chosen at random as part of a class assessment. Use the finite population correction to compute the standard deviation of the mean of the 20 exams. 2.01 b. In general, is the standard deviation computed with the correction smaller or larger than the standard deviation computed without it? smaller c. Use the finite population correction to show that if all 100 exams are sampled, the standard deviation of the sample mean is 0. Explain why this is so. Answers to Check Your Understanding Exercises for Section 7.3 1. ��̄ x = 6, ��̄ x = 0.8 2. ��̄ x = 17, ��̄ x = 2.0 3. a. 0.1894 Tech: 0.1884 b. The probability that ̄x is less than 8 is 0.0384 Tech: 0.0385. If we define an event whose probability is less than 0.05 as unusual, then this is unusual. 4. a. 0.6449 b. The probability that ̄x is greater than 48 is 0.3372 Tech: 0.3373. This event is not unusual. SECTION 7.4 The Central Limit Theorem for Proportions Objectives 1. Construct the sampling distribution for a sample proportion 2. Use the Central Limit Theorem to compute probabilities for sample proportions A computer retailer wants to estimate the proportion of people in her city who own laptop computers. She cannot survey everyone in the city, so she draws a sample of 100 people and surveys them. It turns out that 35 out of the 100 people in the sample own laptops. The proportion 35∕100 is called the sample proportion and is denoted ̂p. The proportion of people in the entire population who own laptops is called the population proportion and is denoted p. DEFINITION In a population, the proportion who have a certain characteristic is called the population proportion. In a simple random sample of n individuals, let x be the number in the sample who have the characteristic. The sample proportion is ̂p = x n Notation: ∙ The population proportion is denoted by p. ∙ The sample proportion is denoted by ̂p. If several samples are drawn from a population, they are likely to have different values for ̂p. Because the value of ̂p varies each time a sample is drawn, ̂p is a random variable, and it has a probability distribution. The probability distribution of ̂p is called the sampling distribution of ̂p. Objective 1 Construct the sampling distribution for a sample proportion An Example of a Sampling Distribution To present an example, consider tossing a fair coin five times. This produces a sample of size n = 5, where each item in the sample is either a head or a tail. The proportion of times the coin lands heads will be the sample proportion ̂p. Because the coin is fair, the probability that it lands heads each time is 0.5. Therefore, the population proportion
navidi_monk_elementary_statistics_2e_ch7-9
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